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MATH232 – Mathematical Techniques

2014 – S2 Day

General Information

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Unit convenor and teaching staff Unit convenor and teaching staff Lecturer
Elena Vynogradova
Contact via elena.vynogradova@mq.edu.au
E7A 204
Unit Convenor
Stuart Hawkins
Contact via stuart.hawkins@mq.edu.au
E7A 212
Credit points Credit points
3
Prerequisites Prerequisites
MATH133 or MATH136
Corequisites Corequisites
Co-badged status Co-badged status
Unit description Unit description
This unit develops techniques and skills that are fundamental in the study and application of mathematics at an advanced level. In any successful application, two contrasting but complementary skills must be developed: the ability to formulate a given real-world problem in appropriate mathematical terms; and sufficient knowledge to obtain useful information and testable predictions from that model, by analytical and numerical means. The unit shows how differential equations arise as mathematical models of such real phenomena in science, engineering and the social sciences, and introduces some tools – Fourier series and numerical methods – for the study and eventual solution of these equations. Fourier series and transforms are particularly useful in those situations where the system response (and indeed many functions) can be seen as a complex sum of simpler vibrations or oscillations. When analytical methods fail, or provide only limited information about the model, numerical techniques are essential to quantify its behaviour precisely; some simple methods are introduced and the conditions under which reliable and accurate solutions may be obtained are described.

Important Academic Dates

Information about important academic dates including deadlines for withdrawing from units are available at https://students.mq.edu.au/important-dates

Learning Outcomes

  1. Demonstrate knowledge of the principles and concepts of mathematical modelling and Fourier series.
  2. Demonstrate understanding of the role of mathematics in other disciplines and the way other disciplines contribute to the development of the mathematical sciences.
  3. Demonstrate the ability to formulate and model practical and abstract problems in mathematical and/or statistical terms using a variety of methods.
  4. Apply mathematical and/or statistical principles, concepts, techniques and technology to solve practical and abstract problems.

Assessment Tasks

Name Weighting Due
Five assignments 20% Week 4, 6, 9, 11, 13
One Test 20% Week 7
Final examination 60% University Examination Period

Five assignments

Due: Week 4, 6, 9, 11, 13
Weighting: 20%

-
This Assessment Task relates to the following Learning Outcomes:
  • Demonstrate knowledge of the principles and concepts of mathematical modelling and Fourier series.
  • Demonstrate understanding of the role of mathematics in other disciplines and the way other disciplines contribute to the development of the mathematical sciences.
  • Demonstrate the ability to formulate and model practical and abstract problems in mathematical and/or statistical terms using a variety of methods.
  • Apply mathematical and/or statistical principles, concepts, techniques and technology to solve practical and abstract problems.

One Test

Due: Week 7
Weighting: 20%

-
This Assessment Task relates to the following Learning Outcomes:
  • Demonstrate knowledge of the principles and concepts of mathematical modelling and Fourier series.
  • Demonstrate understanding of the role of mathematics in other disciplines and the way other disciplines contribute to the development of the mathematical sciences.
  • Demonstrate the ability to formulate and model practical and abstract problems in mathematical and/or statistical terms using a variety of methods.
  • Apply mathematical and/or statistical principles, concepts, techniques and technology to solve practical and abstract problems.

Final examination

Due: University Examination Period
Weighting: 60%

-
This Assessment Task relates to the following Learning Outcomes:
  • Demonstrate knowledge of the principles and concepts of mathematical modelling and Fourier series.
  • Demonstrate understanding of the role of mathematics in other disciplines and the way other disciplines contribute to the development of the mathematical sciences.
  • Demonstrate the ability to formulate and model practical and abstract problems in mathematical and/or statistical terms using a variety of methods.
  • Apply mathematical and/or statistical principles, concepts, techniques and technology to solve practical and abstract problems.

Delivery and Resources

Classes

Lectures: you should attend two hours of each lecture stream each week, making a total of four hours.

Tutorials: you should attend one tutorial each week.

Required and Recommended Texts and/or Materials

The required text for MATH232 is available for download on

You should download and study these.

The online notes are intended primarily as a source of reference. These are not intended to be treated as the only source for learning.

The following texts provide useful references for various sections of the course:

    • A. Pinkus & S. Zafrany, Fourier Series and Integral Transforms, Cambridge University Press, 1997. (QA404.P56 1997)
    • E. Kreyszig, Advanced Engineering Mathematics, Wiley (several editions) (QA401.K7)
    • F. B. Hildebrand, Advanced Calculus for Applications, Prentice-Hall, 1962 (QA303.H55)
    • R. Knobel, R. An introduction to the mathematical theory of waves, American Mathematical Society, 1999 (QA927.K963/1999).
    • G. Fulford, P. Forrester & A. Jones, Modelling with Differential and Difference Equations/i>, Cambridge University Press, 1997 (QA401.F85)
  • G. D. Smith, Numerical Solution of Partial Differential Equations, Oxford University Press, 2nd /3rd Edition (QA374.S56/1978, 1985)
  • K.K. Tung, Topics in Mathematical Modelling, Princeton University Press, 2007.

Other similar texts are available in the Library.

Technology Used and Required

Students are expected to have access to an internet enabled computer with a web browser and Adobe Reader software. Several areas of the university provide wireless access for portable computers. There are computers for student use in the Library and in the Numeracy Centre (C5A 255).

Difficulties with your home computer or internet connection do not constitute a reasonable excuse for lateness of, or failure to submit, assessment tasks.

Unit Schedule

WEEK

A

 

TASK DUE

1

Fourier series

Introduction to modelling

 

2

Fourier series (ctd.). Bessel's inequality.

Modelling with ordinary differential equations (ODEs)

 

3

Fourier series: convergence

Models drawn from biology, physics and other fields

 

4

Fourier series: differentiation

 

Assignment 1

5

Fourier series: integration

   

6

Fourier series: application to ODEs and PDEs

ODEs and phase plane

Assignment 2

7

Fourier series: application to ODEs and PDEs (ctd.)

Modelling with PDEs

 Test 1
 

MID-SEMESTER BREAK

   

8

Fourier transforms

Heat and diffusion: PDE models

 

9

Fourier transforms (ctd.)

Waves and potentials: PDE models

Assignment 3

10

Fourier transforms: application to differential equations

Modelling with maps

 

11

Laplace transforms, convolutions

The logistic map: period doubling and chaos

Assignment 4

12

Application to ODEs

Brief introduction to numerical methods for ODEs

 

13

Revision

Revision

Assignment 5

Policies and Procedures

Macquarie University policies and procedures are accessible from Policy Central. Students should be aware of the following policies in particular with regard to Learning and Teaching:

Academic Honesty Policy http://mq.edu.au/policy/docs/academic_honesty/policy.html

Assessment Policy  http://mq.edu.au/policy/docs/assessment/policy.html

Grading Policy http://mq.edu.au/policy/docs/grading/policy.html

Grade Appeal Policy http://mq.edu.au/policy/docs/gradeappeal/policy.html

Grievance Management Policy http://mq.edu.au/policy/docs/grievance_management/policy.html

Disruption to Studies Policy http://www.mq.edu.au/policy/docs/disruption_studies/policy.html The Disruption to Studies Policy is effective from March 3 2014 and replaces the Special Consideration Policy.

In addition, a number of other policies can be found in the Learning and Teaching Category of Policy Central.

Student Code of Conduct

Macquarie University students have a responsibility to be familiar with the Student Code of Conduct: https://students.mq.edu.au/support/student_conduct/

Student Support

Macquarie University provides a range of support services for students. For details, visit http://students.mq.edu.au/support/

Learning Skills

Learning Skills (mq.edu.au/learningskills) provides academic writing resources and study strategies to improve your marks and take control of your study.

Student Enquiry Service

For all student enquiries, visit Student Connect at ask.mq.edu.au

Equity Support

Students with a disability are encouraged to contact the Disability Service who can provide appropriate help with any issues that arise during their studies.

IT Help

For help with University computer systems and technology, visit http://informatics.mq.edu.au/help/

When using the University's IT, you must adhere to the Acceptable Use Policy. The policy applies to all who connect to the MQ network including students.

Graduate Capabilities

Discipline Specific Knowledge and Skills

Our graduates will take with them the intellectual development, depth and breadth of knowledge, scholarly understanding, and specific subject content in their chosen fields to make them competent and confident in their subject or profession. They will be able to demonstrate, where relevant, professional technical competence and meet professional standards. They will be able to articulate the structure of knowledge of their discipline, be able to adapt discipline-specific knowledge to novel situations, and be able to contribute from their discipline to inter-disciplinary solutions to problems.

This graduate capability is supported by:

Learning outcomes

  • Demonstrate knowledge of the principles and concepts of mathematical modelling and Fourier series.
  • Demonstrate understanding of the role of mathematics in other disciplines and the way other disciplines contribute to the development of the mathematical sciences.
  • Demonstrate the ability to formulate and model practical and abstract problems in mathematical and/or statistical terms using a variety of methods.
  • Apply mathematical and/or statistical principles, concepts, techniques and technology to solve practical and abstract problems.

Assessment tasks

  • Five assignments
  • One Test
  • Final examination

Problem Solving and Research Capability

Our graduates should be capable of researching; of analysing, and interpreting and assessing data and information in various forms; of drawing connections across fields of knowledge; and they should be able to relate their knowledge to complex situations at work or in the world, in order to diagnose and solve problems. We want them to have the confidence to take the initiative in doing so, within an awareness of their own limitations.

This graduate capability is supported by:

Learning outcomes

  • Demonstrate knowledge of the principles and concepts of mathematical modelling and Fourier series.
  • Demonstrate understanding of the role of mathematics in other disciplines and the way other disciplines contribute to the development of the mathematical sciences.
  • Demonstrate the ability to formulate and model practical and abstract problems in mathematical and/or statistical terms using a variety of methods.
  • Apply mathematical and/or statistical principles, concepts, techniques and technology to solve practical and abstract problems.

Assessment tasks

  • Five assignments
  • One Test
  • Final examination

Effective Communication

We want to develop in our students the ability to communicate and convey their views in forms effective with different audiences. We want our graduates to take with them the capability to read, listen, question, gather and evaluate information resources in a variety of formats, assess, write clearly, speak effectively, and to use visual communication and communication technologies as appropriate.

This graduate capability is supported by:

Learning outcomes

  • Demonstrate knowledge of the principles and concepts of mathematical modelling and Fourier series.
  • Demonstrate understanding of the role of mathematics in other disciplines and the way other disciplines contribute to the development of the mathematical sciences.
  • Demonstrate the ability to formulate and model practical and abstract problems in mathematical and/or statistical terms using a variety of methods.
  • Apply mathematical and/or statistical principles, concepts, techniques and technology to solve practical and abstract problems.

Assessment tasks

  • Five assignments
  • One Test
  • Final examination

Capable of Professional and Personal Judgement and Initiative

We want our graduates to have emotional intelligence and sound interpersonal skills and to demonstrate discernment and common sense in their professional and personal judgement. They will exercise initiative as needed. They will be capable of risk assessment, and be able to handle ambiguity and complexity, enabling them to be adaptable in diverse and changing environments.

This graduate capability is supported by:

Learning outcomes

  • Demonstrate knowledge of the principles and concepts of mathematical modelling and Fourier series.
  • Demonstrate understanding of the role of mathematics in other disciplines and the way other disciplines contribute to the development of the mathematical sciences.
  • Demonstrate the ability to formulate and model practical and abstract problems in mathematical and/or statistical terms using a variety of methods.
  • Apply mathematical and/or statistical principles, concepts, techniques and technology to solve practical and abstract problems.

Assessment tasks

  • Five assignments
  • One Test
  • Final examination

Critical, Analytical and Integrative Thinking

We want our graduates to be capable of reasoning, questioning and analysing, and to integrate and synthesise learning and knowledge from a range of sources and environments; to be able to critique constraints, assumptions and limitations; to be able to think independently and systemically in relation to scholarly activity, in the workplace, and in the world. We want them to have a level of scientific and information technology literacy.

This graduate capability is supported by:

Learning outcomes

  • Demonstrate knowledge of the principles and concepts of mathematical modelling and Fourier series.
  • Demonstrate understanding of the role of mathematics in other disciplines and the way other disciplines contribute to the development of the mathematical sciences.
  • Demonstrate the ability to formulate and model practical and abstract problems in mathematical and/or statistical terms using a variety of methods.
  • Apply mathematical and/or statistical principles, concepts, techniques and technology to solve practical and abstract problems.

Assessment tasks

  • Five assignments
  • One Test
  • Final examination

Creative and Innovative

Our graduates will also be capable of creative thinking and of creating knowledge. They will be imaginative and open to experience and capable of innovation at work and in the community. We want them to be engaged in applying their critical, creative thinking.

This graduate capability is supported by:

Learning outcomes

  • Demonstrate knowledge of the principles and concepts of mathematical modelling and Fourier series.
  • Demonstrate understanding of the role of mathematics in other disciplines and the way other disciplines contribute to the development of the mathematical sciences.
  • Demonstrate the ability to formulate and model practical and abstract problems in mathematical and/or statistical terms using a variety of methods.
  • Apply mathematical and/or statistical principles, concepts, techniques and technology to solve practical and abstract problems.

Assessment tasks

  • Five assignments
  • One Test
  • Final examination

Engaged and Ethical Local and Global citizens

As local citizens our graduates will be aware of indigenous perspectives and of the nation's historical context. They will be engaged with the challenges of contemporary society and with knowledge and ideas. We want our graduates to have respect for diversity, to be open-minded, sensitive to others and inclusive, and to be open to other cultures and perspectives: they should have a level of cultural literacy. Our graduates should be aware of disadvantage and social justice, and be willing to participate to help create a wiser and better society.

This graduate capability is supported by:

Learning outcomes

  • Demonstrate understanding of the role of mathematics in other disciplines and the way other disciplines contribute to the development of the mathematical sciences.
  • Apply mathematical and/or statistical principles, concepts, techniques and technology to solve practical and abstract problems.

Assessment task

  • Five assignments

Socially and Environmentally Active and Responsible

We want our graduates to be aware of and have respect for self and others; to be able to work with others as a leader and a team player; to have a sense of connectedness with others and country; and to have a sense of mutual obligation. Our graduates should be informed and active participants in moving society towards sustainability.

This graduate capability is supported by:

Learning outcome

  • Demonstrate understanding of the role of mathematics in other disciplines and the way other disciplines contribute to the development of the mathematical sciences.

Assessment task

  • Five assignments

Commitment to Continuous Learning

Our graduates will have enquiring minds and a literate curiosity which will lead them to pursue knowledge for its own sake. They will continue to pursue learning in their careers and as they participate in the world. They will be capable of reflecting on their experiences and relationships with others and the environment, learning from them, and growing - personally, professionally and socially.

This graduate capability is supported by:

Learning outcomes

  • Demonstrate knowledge of the principles and concepts of mathematical modelling and Fourier series.
  • Demonstrate understanding of the role of mathematics in other disciplines and the way other disciplines contribute to the development of the mathematical sciences.
  • Demonstrate the ability to formulate and model practical and abstract problems in mathematical and/or statistical terms using a variety of methods.
  • Apply mathematical and/or statistical principles, concepts, techniques and technology to solve practical and abstract problems.

Assessment tasks

  • Five assignments
  • Final examination

Extra requirements

Satisfactory performance on supervised assessment tasks, such as tests and the final exam, is necessary to pass this unit. If there is a significant difference between a student's marks on supervised assessment tasks and on unsupervised assessment tasks, the scaling of these tasks may be adjusted when determining the final grade, to reflect more appropriately that student's performance on supervised tasks.